how many times does the graph of $4x = 32 - x^2$ cross the $x$-axis?

0
1
2

what are the solutions to the equation?

-2, 16
2, -16
-4, 8
4, -8

Question

how many times does the graph of $4x = 32 - x^2$ cross the $x$-axis?

0
1
2

what are the solutions to the equation?

-2, 16
2, -16
-4, 8
4, -8

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Zeros of Quadratic Functions", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Zeros of Quadratic Functions", "Solving Quadratic Equations" ] } ## Rearrange the quadratic equation into standard form $$ \begin{aligned} &4x = 32 - x^2 \ &x^2 + 4x - 32 = 0 \end{aligned} $$ ## Solve the quadratic equation by factoring $$ \begin{aligned} &(x + 8)(x - 4) = 0 \ &x = -8 \quad \text{or} \quad x = 4 \end{aligned} $$ ## Determine the number of x-intercepts $$ \begin{aligned} &\text{Since there are two distinct real solutions } x = 4 \text{ and } x = -8,\ &\text{the graph crosses the } x\text{-axis exactly } 2 \text{ times.} \end{aligned} $$ ## Question 1 0

1

2 ## Question 2 The solutions to the equation are $4, -8$. { "elements": [ { "type": "functiongraph", "params": [ { "js": "32 - x*x - 4*x", "latex": "y = -x^2 - 4x + 32" }, -10, 6 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = 32 - x^2 - 4x", "withLabel": false } }, { "type": "point", "params": [[4, 0]], "properties": { "name": "(4, 0)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[-8, 0]], "properties": { "name": "(-8, 0)", "size": 4, "color": "#F2557F", "withLabel": true } } ], "timestamps": [0.5, 1.0] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Zeros of Quadratic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Rearrange the quadratic equation into standard form

Solve the quadratic equation by factoring

Determine the number of x-intercepts

Answer:

Question 1

Question 2